Question

the solid shown has a volume of $\frac{c}{3}\pi$ cubic units, where $c$ is a constant. what is the value of $c$?

Explanation:

Step1: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \(V_{cylinder}=\pi r^{2}h\). Given \(r = 2\) (since \(d = 4\), \(r=\frac{d}{2}\)) and \(h = 3\).

$$V_{cylinder}=\pi\times(2)^{2}\times3=12\pi$$

Step2: Calculate the volume of the cone

The formula for the volume of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). Here \(r = 2\) and \(h = 2\).

$$V_{cone}=\frac{1}{3}\pi\times(2)^{2}\times2=\frac{8}{3}\pi$$

Step3: Calculate the total volume

The total volume \(V = V_{cylinder}+V_{cone}\).

$$V=12\pi+\frac{8}{3}\pi=\frac{36\pi + 8\pi}{3}=\frac{44\pi}{3}$$

Since \(V=\frac{c}{3}\pi\), then \(c = 44\).

Answer:

\(44\)